If I have Svellsongur attached to The Ball, do I get two separate instances of the move trigger?

It depends — but if The Ball is currently a unit gear (made one by Lady of Clockwork), yes: you get two separate "When you play or move this" triggers, one original and one from the copy.

The Ball is a Gear, not a Unit, so Svellsongur's Equip can only attach after Orianna's legend ability has turned The Ball into a unit gear (Rule 818.1.b). Once attached, Svellsongur copies The Ball's printed text onto its own Effect Text, which is appended to The Ball's Rules Text (Rule 136.2.c) — so The Ball now has two instances of "choose one — Draw 1. Deal 2 to enemy units here."

They are two independent triggers, not a doubled effect. Both go on the chain when The Ball moves, you pick their order (Rule 383.3.d), and each one makes its own "choose one" choice — so you could Draw 1 twice, Deal 2 twice to enemy units there, or split them. Same as the Hwei/Svellsongur precedent. Doubling the copy doesn't copy a "choose one" into a single bigger choice.